I am able to rent a violin for my youngest. I won't invest in one until she is a big bigger (soze wise, not age wise). It's pretty cheap to rent. We also got our piano for free when we put some ads out. You could try Freecycle or even the newspaper. I am alsways surprised by how many free painos there are. (You probably won't get a great one, and you will have to move it, but you can probably get something decent for a beginner. My kids played on a free paino for years. It was imperfect, but it was an excellent start.
Some places do group lessons a bit cheaper. Check music schools around your area for that. You can also put the word out-- in poster form etc at music shops where they also teach--that you are willing to barter__ whatever for lessons. You might be able to get a reduced rate, even if you can't get free lessons.
The problem comes in later, when kids love it and get really good. Beginners don't cost so much, but older kids tend to. In time, instruments, lessons etc. Because of this, I am considering letting one of my hsers attend music classes at school for additional instruction. (We can do extra curic-- music, sports, art etc) in our town without having to enroll the child for a full school day.
Nice teens in your area might also be a good resource. They wouldn't charge as much. One of mine has helped younger kids.